“A long time ago an older and well known number theorist made some disparaging remarks about Paul Erdos’ work. … I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos’ work could be reduced to a few tricks which Erdos repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over.”

High-level thinkers are not dark magicians — they’re smart-but-otherwise-ordinary people who’ve picked up some interesting tricks.

In this essay, I want to dissolve some of the mysticism that surrounds high-level thinking.

The Wrong Wetware

It’s a common belief that ideas from, say, quantum physics or behavioral neuroscience are headache-inducing and hard to understand. This is true, but it took me years to realize its true for everyone — this stuff is supposed to be hard to understand, even for experts.

Scientists are just as bad as you or me when dealing with abstract concepts. Here’s Australian theoretical physicist Michael Nielsin in Extreme Thinking:

“…most scientists aren’t much better than average at when it comes to intuitively working with such abstractions. We’re stuck with the same equipment as everybody else. That, you see, is the special challenge that faces the scientist: using equipment not cut out for understanding abstract concepts to understand a Universe whose fundamental laws are abstract concepts.”

By “equipment”, Nielsen is talking about what hackers sometimes call “wetware” — our brain and the other “wet” parts of our biology that we use to understand and interact with the world.

It’s easy to for me to forget that humankind is still really, really young. Brutus gutted Caesar with a knife only 60–70 generations ago, and it’s probably been only ~300 generations since our nomadic ancestors decided to stop chasing deer and settle down to grow crops.

In other words, our bodies aren’t ‘designed’ for modern life at all.

We humans didn’t evolve to think about cryptocurrency, orthogonal matrices, or interplanetary space travel, which means this stuff is really, really hard for us:

“…there is no reason to suppose the laws of Nature are particularly accessible to the human mind. The mind developed as a way of furthering competition with our fellow living beings, and only incidentally to understand the laws of the Universe. It’s good at recognizing faces, making tools, and understanding social situations. It’s not good at solving mathematical equations or other abstract reasoning. In doing science we are borrowing equipment developed for another purpose, and we should not be surprised if sometimes the equipment is not cut out for the task.”

Here’s a simple example of how hard some ideas are to understand: Try to imagine a six-dimensional space. Tried? Okay, well it’s impossible. It’s not just impossible for you — it’s impossible for everyone, including theoretical physicists.

But theoretical physicists do think in six dimensions all the time. So how do they do it?

Jerry-Rigging the Mind

I wish we learned about this evolution-modernity mismatch in school, but I didn’t. The first book to clue me in on it was Harvard linguist Steven Pinker’s book The Blank Slate:

“Some of our perplexities may come from a mismatch between the purposes for which our cognitive faculties evolved and the purposes to which we put them today. … Our minds keep us in touch with aspects of reality — such as objects, animals, and people — that our ancestors dealt with for millions of years. But as science and technology open up new and hidden worlds, our untutored intuitions may find themselves at sea.”

“…we have no mental tools to grasp [these subjects] intuitively. We depend on analogies that press an old mental faculty into service, or on jerry-built mental contraptions that wire together bits and pieces of other faculties. Understanding in these domains is likely to be uneven, shallow, and contaminated by primitive intuitions.”

It’s Metaphors All the Way Down

Remember metaphors? If you’re like me, you heard your English teacher say stuff like “Laughter is the music of the soul” or “His heart is a cold iron” and thought, “Nope, no thank you. This is kooky, cringey, and completely useless. Time to take a nap.”

But it turns out metaphors aren’t just for English teachers, poets, and singer-songwriters. Metaphors are for everyone, because they are the way we understand anything at all.

“The body … precedes language and abstract thought during individual development. A child first comes to know the world through his or her body, and later developmental steps build on top of that understanding. As George Lakoff argues, our capacity for abstract thought is grounded in (conceptual) metaphor. We reason about abstract domains in terms of more concrete, embodied domains. ‘The very words which form the building blocks of explicit thought are themselves all originally metaphors, grounded in the human body and its experience,’ writes Iain McGilchrist, channeling Lakoff.”

Basically, we understand ideas in terms of other ideas. To understand an abstract concept like “love” we need a bunch of metaphors like JOURNEY, MADNESS, WAR, and so on.

But, if you think about it, our big love-web can’t just be a mass of interwoven metaphors floating in outer space. Metaphors have to come from somewhere.

Well, this “somewhere” happens to be our bodies.

Bodily experience is all we have, and we have to use these experiences to try and understand difficult, abstract ideas. Humans operate in three-dimensions — this makes six-dimensional thinking pretty hard.

The answer — as Pinker pointed out earlier — is to (a) using analogies [which come from experience] and wire together “Jerry-built mental contraptions”. Turns out, we can use a whole bunch of metaphors together to help understand difficult, abstract ideas.

Here’s a high-quality, high-effort graphic to help express this:

This reminds me of that one story. You know, the one where a bunch of kinky Chinese blind men are all rubbing up on an elephant, trying to guess what they are touching:

If they communicate well, they might get a pretty good idea of what the elephant looks like, but never a perfect one.

It gets worse. The more complex and abstract an idea is, the more metaphors we’ll need to “stack” on top of each other to understand it. This is probably why quantum mechanics is so damn hard to understand — it’s really, really far from our natural, embodied, caveman way of understanding the world.

To stretch the analogy, imagine if the team of blind men went next door to the School for the Deaf and tried to communicate (via sign language, I guess) what it was that they had “rubbed up on.” It’d be pretty hard.

Anyway, another way to help master abstract ideas is to give them a sexy name. Let’s do that now. I’m gonna call this metaphorical scaffolding — you can ‘stack’ metaphors on top of one another to help ‘reach’ an understanding of difficult ideas.

Sounds pretty sexy to me… but I’m Asian

Back to Practice

Okay, enough theory. Let’s switch to some field work — how does metaphorical scaffolding work in practice?

Let me share a quick example from a paper in the American Journal of Physics. The authors try to show how amateur students think differently from their professors.

Teasing Students

First, let’s explore (read: make fun of) how amateur students try to solve problems. Unsurprisingly, most students come out of college physics just as useless as when they went in:

“Students, at the end of their conventional study, have little structure to their knowledge. Their understanding consists of random facts and equations that have little conceptual meaning. When given a problem, they identify some structural feature described in the problem (an inclined plane, a rope, a spring, etc.). They then search randomly for and inappropriately use an equation they associate with that feature.”

This is the classic “plug and chug” problem solving that we learned to do in school. Any small shift in context destroys your ability to think — this is the opposite of what we want.

So how do pro physicists do it differently?

How Pro Physicists Think

Remember our sexy word ‘metaphorical scaffolding’? Well keep that in mind as you read this:

“A physicist depends on qualitative analysis and representations [metaphors! intuition! visualization! banzai!] to understand and help construct a mathematical representation of a physical process. … The diagram serves several purposes. (1) It summarizes the prominent features of a process while removing noisy details that distract from understanding — in short, the diagram contributes to understanding and to physical intuition. (2) Diagrams can be strung together to reason qualitatively about more complex processes [metaphorical scaffolding!!!]. (3) Using special rules and heuristics, the diagram can be used to construct a detailed mathematical representation of the process.”

This was so, so cool when I read it. Turns out, physicists are using metaphorical scaffolding to understand and solve problems:

For more complex ideas, string together multiple diagrams into a ‘scaffolding’ to help yourself understand

Then, lastly, translate this diagram into mathematics to get an answer

Here’s an example comparing a diagram to its mathematical representation:

You can know zero math and still appreciate that the diagram is much better organized and easier to understand.

A Last Hoorah

For years and years, I thought relying on diagrams & intuition was a weakness, a crutch for the not-smart-enough. I was wrong. We are all humans, and all humans need intuition to understand difficult ideas.

Metaphorical scaffolding seems like a great way to get there.

The teachers I admire today (like the physicists above) all start with intuition & metaphor before layering in rigor & abstraction. I try to do the same in my writing. If you’ve reached the end of this 1900-word essay, I guess I’ve done a half-decent job at making the ideas clear, digestible, and (gasp) even fun to read.

I’ll never forget these words from the professor-turned-novelist Umberto Eco: “It’s only publishers and some journalists who believe that people want simple things. People are tired of simple things. They want to be challenged.”

But, as is only proper for an essay full of physicists, let me end with another quote from the great Richard Feynman:

“We are at the very beginning of time for the human race. It is not unreasonable that we grapple with problems. But there are tens of thousands of years in the future. Our responsibility is to do what we can, learn what we can, improve the solutions, and pass them on.”